An analog computer for solving sets of simultaneous relations

ABSTRACT

An analog computer having a multiplexed negative feedback system comprising : an electrical network of impedances defining linear functions and which network is reciprocal between first and second sets of conductors of the network; and signal transmitting-receiving devices connected to said sets and which multiplex signals passing through the network between said sets in opposite directions. Solutions of the functions can be determined from the signals passing through the network in one direction from the transmitting-receiving devices connected to one of said sets when the system achieves equilibrium, and the signals passing through the network in the opposite direction from the other transmitting-receiving devices being negative feedback signals for controlling said devices connected to said one set of conductors. The multiplexing can be frequency or time multiplexing.

United States Patent [191 Walton 1 1 ANALOG COMPUTER FOR SOLVING SETS OF SIMULTANEOUS RELATIONS [76] Inventor: John Hugh Davey Walton, 2 Broadfield Road, Kent, England [22] Filed: March 9, 1971 [21] Appl. No.: 122,338

[52] US. Cl. ..235/l80, 235/184, 235/185 [51] Int. Cl. .;...G06g 7/34 [58] Field of Search ..235/l80, 184, 185

[56] References Cited UNITED STATES PATENTS 2,543,650 2/1951 Walker ..235/180 2,740,584 4/1956 Jacobi et a1. 235/180 2,808,989 10/1957 Younkin 235/180 3,038,660 6/1962 Honnell et al. 235/180 3,188,454 6/1965 Honore et a1 235/180 3,443,078 5/1969 Noronha et al.... ....235/180 3,517,169 6/1970 Malavard'et al ..235/184 FOREIGN PATENTS OR APPLlCATlONS 1/1964 Great Britain ..235/180 1 Feb. 6, 1973 Primary Examiner-Felix D. Gruber Attorney-Cushman, Darby & Cushman [57] ABSTRACT An analog computer having a multiplexed negative feedback system comprising an electrical network of impedances defining linear functions and which network is reciprocal between first and second sets of conductors of the network; and signal transmitting- .receiving devices connected to said sets and which multiplex signals passing through the network between I said sets in opposite directions. Solutions of the functions can be determined from the signals passing through the network in one direction from the transmitting-receiving devices connected to one of said sets when the system achieves equilibrium, and the signals passing through the network in the opposite direction from the other transmitting-receiving devices being negative feedback signals for controlling'said devices connected to said one set of conductors. The mul tiplexing can be frequency or time multiplexing.

21 Claims, 8 Drawing Figures SHEET 1 or 6 mw g Q gunk, T a

SHEET 2 OF 6 PATENTED FEB 6 I973 PATENTED F EH 6 I975 SHEET 3 BF 6 ANALOG COMPUTER FOR SOLVING SETS OF SIMULTANEOUS RELATIONS BACKGROUND OF THE INVENTION This invention relates to analog computers.

Problems arising in the control of industrial manufacturing processes, in the formulation of business and economic strategies and in the optimization or solution of equations and inequalities can often be resolved by the application of analog computing techniques.

The conditions or restraints of such problems can be represented by the parameters of an electrical network.

7 Such a network could comprise an array of intercon nected electrical components which connect together, either directly or indirectly, a plurality of electrical terminals. Electrical generating means and measuring means could be connected to said terminals, the application of voltage to one or more of those terminals causing current to flow in the network and voltages to appear at other terminals. The values of the voltages appearing at said other terminals would be dependent on both the configuration of the interconnections in the network and on the values of the electrical components of which the network is composed.

One important embodiment of such a network is one described in BritishvPat. Specification No. 1,173,900 and is one in which resistance elements are arranged, figuratively speaking, in matrix form to interconnect at least some of the crossing points of row and column" conductors, the terminals of a first set of terminals being connected to respective ones of the row conductors and the terminals of a second set of terminals being connected to respective ones of column conductors. In this network, a signal applied at any terminal of one of said sets of terminals will produce signals at at least one of the terminals of the other set of terminals. The network could be utilized in the following manner to solve a set of linear simultaneous equations: I

The columns (or rows, according to the network of British Pat. No. 1,173,900 if so desired) of resistance elements represent respective ones of the set of equations and the resistance elements in a column represent respective terms of the equation represented by that column. The value of each resistance element is inversely proportional to the constant numerical coefficient, or the constant numerical value, of the term which it represents. Resistance elements arranged in the same rowhave the same variable voltage applied to them through the associated terminal of said first set of terminals, that voltage representing the unknown variable common to each term represented by a resistance element in that row. The number of variables in the equa- I ble to adjust the variable voltages to such values that the voltages appearing at the second set of terminals are all equal to zero. When such a state exists, all the equations of the set are simultaneously satisfied and their solutions can be determined by measurement of the variable voltages.

In addition to the above-described solution of simultaneous equations, it is also possible, utilizing this embodiment of network, to determine a system of linear inequalities or even a system comprising both linear inequalities and one or more simultaneous equations. In these cases, where a column represents an inequality, there is stipulated for that terminal of the second set of terminals which is associated with that column not a zero voltage condition but a condition that, for the inequality to be satisfied, the voltage appearing at the terminal must be of a predetermined polarity or phase. Linear inequalities may occur in the use of the mathematical technique known as Linear Programming and may be employed to describe, for example, problems involving the allocation of available supplies of commodities to satisfy various specific demands for those commodities.

It will be appreciated that whenever a large number of equations or inequalities containing a correspondingly large number of independent variables are to be solved together as has been described hereinbefore, the number of rows and columns of resistance elements, the number of voltage sources required to apply the variable voltages to the first set of terminals and the number of meters required to indicate the voltages at the second row of terminals will necessarily be large also. Manual adjustment of the variable voltage sources and simultaneous visual monitoring of all the meters would prove to be a very exacting, if not impossible, task for a human computer operator. It would be possible to ease the operator's task by coupling each terminal of the second set of terminals in common to an electric lamp such that whenever one of the equations or inequalities is not satisfied the lamp is illuminated. This would avoid the necessity of the operator having to scan the meters in order to determine when and if a condition, of equality or inequality was no longer satisfied. Notwithstanding the provision of the electric lamp, however, the operator on sensing illumination of the lamp would still need to scan the meters to determine which of the particular conditions had been broken. Moreover, it would still be necessary to perfonn the tedious manual adjustment of the variable voltage sources. I

It is an object of the present invention to reduce the amount of operator intervention required.

SUMMARY OF THE INVENTION According to the present invention, there is provided, in an analog computer comprising an electrical network having a first set of conductors, a second set of conductors and electrical impedance elements coupling said first set of conductors tosaid second set of conductors such that a signal applied at any one ofv said conductors of one of said sets will produce at at least one of said conductors ofthe other of said sets signals having values determined by those of said impedance elements which couple said one conductor to said other set.

This includes a negative feedback system having a 1 I plurality of transmitting-receiving devices each having an output and an input coupled in common to an associated one of said conductors of said first and second sets and each being operable to apply to its output an output signal varying in dependence upon an input signal received at its input.

The negative feedback system also includes multiplexing means for multiplexing'the signals which pass from said first set of conductors to said second set of conductors with the signals which pass from said second set of conductors to said first set of conductors.

The wholeassernbly of network and transmittingreceiving devices can be designed to constitute a stable system with multiple negative feedback, so that the system will reach equilibrium just as if the manual adjustment already described had been carried out.

In a first embodiment, frequency multiplexing is employed. In this case frequency conversion means provide the signals'passing through the network from said first set of conductors to said second set of conductors at a first frequency'and said signals which pass from said second set of conductors to said first set of conductors at a second frequency. The frequency conversion means may include modulating means.

Preferably, said first frequency is zero, in which case the frequency conversion means providing those of said signals which are at zero frequency comprise de-modulating means and said' frequency conversion means providing said signals at said second frequency include modulating means.

In a second embodiment, time multiplexing is employed. In this case, signals pass through the network from the first to the second set of terminals during a first set of periods of time and from the second to the first set of terminals during a second set of periods of time The periods of the first set alternate with and are distinct from the periods of the second set.

Time multiplexing can be achieved utilizing two clock pulse trains. The particular form of transmittingreceiving device used in time multiplexing may be such that it can be switched from the transmit to the receive function in response to said pulse trains in order to cause signals to pass through the network in alternate directions as described in the last paragraph.

In a preferred embodiment, which can be adapted either for frequency or time multiplexing, one conductor of the first set of conductors of the network is connected to impedance elements representing constant terms in functions represented by those impedance elements, and others, which are connected to conductors of said second set of conductors. Said one conductor is fed by a further transmitting-receiving device operable to produce an output signal which varies with its input signal without multiplexing between its input and output signals andhaving its output only connected to said one conductor. The input. of the further device is connected to one of the conductors of the second set. Comparison means are provided in the arrangement for comparing the output of said further transmittingreceiving device with the output of a, preferably adjustable, reference source. To implement the comparison means, one may provide a further conductor of said first set and a further conductor of said second set, both of which have a first impedance elementwhich connects them to one another and a second impedance element which connects the further conductor of said second set to said one conductor fed by the further transmitting-receiving device. The reference source is connected to the further conductor of said first set so a comparison signal which will represent the difference between the output signals of the reference source and the further transmitting-receiving device. An additional transmitting-receiving device having its input connected to said further one of the second conductors produces an output signal in dependence upon said difference. The output of the, additional transmittingreceivin-gdevice is not necessarily connected in common to the input-of the device but is connected to that conductor of said second set connected to the input of said further transmitting-receiving device. The output signal of said additional transmitting-receiving device is thereby fed back into the network.

BRIEF DESCRIPTION OF THE DRAWINGS For a better understanding of the invention, and to show how the same may be carried into effect, reference will now be made, by way of example, to the accompanying drawings in which:

FIG. 1 is the circuit diagram of a negative feedback system of an analog computer, the system employing frequency multiplexing and having a network of impedance elements representing a set of simultaneous equations and/or inequalities;

FIG. 2 is the circuit diagram of a preferred embodiment of the negative feedback system of FIG. 1 employing frequency multiplexing and showing resistances as a particular example of the impedance elements of the network; I

FIG. 3 shows a modification of part of one of the transmitting-receiving devices shown in FIG. 2 enabling the device to determine both equations and inequalities in the network;

FIG. 4 shows a voltage-stabilizer circuit for another of the transmitting-receiving devices shown in FIG. 2;

FIG. 5 is the circuit diagram of a further preferred embodiment of a negative feedback system employing frequency multiplexing;

FIG. 6 is the circuit of a negative feedback system employing time multiplexing;

FIG. 7 is the circuit diagram of a modification of the preferred embodiment of FIG. 5 employing time multiplexing; and

FIG. 8 is the circuit diagram of a negative feedback system employing a plurality of impedance networks.

DETAILED DESCRIPTION OF THE DRAWINGS FIG. 1 is the generalized circuit diagram of a negative feedback system in an analog, a preferred embodiment of which shown in FIG. 2 computer. The system can be used for solving sets of simultaneous equations, for determining systems of linear inequalities or for solving problems involving both simultaneous equations and linear inequalities together. The system comprises a reciprocal network 1 having first and second sets of conductors, for example terminals, shown connecting into the left and right hand sides respectively of the network 1. The first and second sets of terminals are connected to each other through the network 1 by connections (shown in dotted lines) including electri cal impedance elements, for example resistances. The

connections are such that a signal applied at anyone of the terminals of one of the sets will produce at at least some of the terminals of the other of the sets signals having values determined by those of the resistances which couple that one terminal to that other set of terminals in such a manner that the network is an electrical analog of a problem to be solved. If it is desired, some of the terminals may be connected to a common grounded busbar 2 when not required. The network 1 is shown merely symbolically in FIG. 1 and is intended to have the form shown for network 18 in FIG. 2. Furthermore, the networks 1 and 18 are similar to the network ABC shown in FIG. 1 of British Pat. No. 1,173,900 which shows network ABC having a plurality of variable potentiometers connected to one set of terminals so that signals representing the trial solutions to a set of simultaneous relations set up on the network can be applied to it and manually adjusted until all the signals at another set of terminals, monitored by a set of measuring instruments, all become equal to zero indicating that the relations are simultaneously satisfied.

In FIG. 1 ofthe present application, at least some of the terminals of the first set of terminals of network 1 are connected to generators 3 of a first type which replace the aforesaid potentiometers of British Pat. No. 1,173,900. At least some of the terminals of the second set are. connected according to the present embodiment to generators 4 of a second type which replace the aforesaid measuring instruments of the above-mentioned British Patent. For the sake of clarity only those generators connected to the lowest terminals of each of the two sets of terminals are shown. The generators 3 apply to their associated network terminals signals at a first frequency and the generators 4 are responsive to signals appearing at their associated network terminals, which signals are at the first frequency and have travelled through the network from the generators 3. Similarly, the generators 4 apply to their associated network terminals signals at a second frequency which travel through the network and control the generators 3 in a negative feed back sense. The generators 3 and 4 respond to the signals at the second and first frequencies respectively so as to vary the magnitude of the signals which they produce in dependence upon the magnitude of the signals they receive at the second and first frequencies respectively. Eventually the system will reach equilibrium with substantially zero value signals appearing at the right-hand terminals of the network 1 and with the simultaneous solutions to the various relations being fed to the left-hand terminals by generators 3. The fact that signals passing in opposite directions through the network 1 are at first and second different frequencies (frequency multiplexing) enables the signals passing in opposite directions not to interfere with each other in respect of their controlling influence on the two sets of generators such as 3 and 4 respectively. 4

The circuitry and operation of generators 3 and 4 will now be described in more detail. Considering generator 3, a terminal 5 is connected via a common input and output of the generator 3 to a frequency filter 6 which prevents interference between output signals leaving the generator at the first frequency and input signals entering the generator at the second frequency. The frequency filter 6 is connected to the input of an amplifier 8, the output of the amplifier 8 being connected to the input of a modulator 9 acting as a frequency conversion means converting the second frequency to the first frequency. A power amplifier 7 applies to the terminal 5, through the filter 6, a signal at the first frequency, the magnitude of the signal being varied by the modulator 9 in dependence upon the magnitude of the input signal of the generator 3. The modulator 9 is controlled by the input signal through the filter 6 and the amplifier 8.

The circuitry and mode of operation of generator 4 is similar to that of generator 3. Input signals at the first frequency are applied via a common input and output of the generator 4 to a frequency filter 13 and control through an amplifier 14 a modulator 11 acting as a frequency conversion means converting the second frequency to the first frequency. A power amplifier 15 supplies to the terminal associated with the generator 4, through the frequency filter 13, output signals at the second frequency which are modulated in dependence upon the value of the input signal. Preferably amplifiers 14 and 15 correspond to amplifiers 7 and 8 respectively in regard to their operating frequencies.

The modulators 9 and 11 are fed with reference signals from alternating current generators l0 and 12 respectively to operate the modulators at the frequencies of the respective alternating current generators. Preferably the generator 10 supplies a reference signal at the first frequency and generator 12 a reference signal at the second frequency. The combination of modulators 9 and 11 of all the generators 3 and 4, together with the alternating current generators 10 and 12, constitute frequency multiplexing means which multiplex the signals passing from left to right through the network 1 with the signals passing from right to left, to prevent them interfering with one another.

Meters 16 and 17 connected between the grounded busbar 2 and the terminals associated with generators 3 and 4 measure the input and output voltages appearing at those terminals at one of the frequencies.

FIG. 2 shows the circuit diagram of a preferred embodiment of the negative feedback system in an analog computer as shown in FIG. 1. The system comprises a network 18 which is similar to the network 1 of FIG. 1 but which is shown and described in more detail.

The network 18 comprises a first set of conductors some of which are shown at 19, 20 and 21 and a second set of conductors some of which are shown at 22, 23 and 24. The first set form rows and the second set columns of an electrical matrix. A plurality of resistance elements 25 are connected between the rows and columns at crossing points of the matrix.

A plurality of simultaneous equations can be solved by the network 18, the equations being represented by respective columns of the network. Each term of an equation is represented by a resistance element in the associated column, the value of each resistance element being inversely proportional to the constant numerical coefficient, or the constant numerical value, of the term which it represents. Resistance elements in the same row have the same signal applied to them, for example by a signal generating means in the form of a variable D.C. source 26 through the row conductor 20, or by a transmitting-receiving device 27 through the row conductor 21. The signal applied to the row conductor 21 represents one of the unknown variables and is common to each of the terms represented by the resistance elements connected to the row conductor 21. Similarly, a signal applied to the row conductor 19 represents that one of the unknown variables which is common to the terms represented by resistance elements connected to the row conductor 19.

The constant signal applied to the row conductor 20, if it is adjusted to be equal to unity, enables resistance elements connected to the row conductor to represent the respective constant terms of the equations. The value of the signals applied'to the conductors 19, 20 and 21 can be monitored on signal indicating means such as meters 28 and 29.

It is possible to vary the signals which are applied to the first set of conductors from transmitting-receiving devices such as device 27 until the output signal appearing at each conductor of the second set of conductors and monitored by signal indicating means such as meters 30, 31 and 32 is equal to zero. When such a state exists, each simultaneous equation represented by a column of the network 18 is satisfied and the solutions of the equations can be read directly from the meters 28 and 29 etc.

In addition to the above-described solution of simultaneous equations, using the network 18, it is also possible to determine a system of linear inequalities or even a system comprising both linear equalities and one or more simultaneous equations. Where a column represents an inequality, there can be stipulated for that conductor of the second set of conductors, for example the conductor 22, a condition that for the inequality to be satisfied the signal appearing at the conductor must be of a predetermined polarity of phase with respect to the device 27 and source 26.

It will be appreciated that the number of equations and/or inequalities which can be simultaneously solved using the network 18 is limited only by the number of available columns in the network. Similarly, the number of variables in those equations and/or inequalities is limited only by the number of available rows.

Transmitting-receiving devices of a first type, such as device 27, are connected to at least some of the row conductors of the first set of conductors of the network 18 to apply to the network an input signal at a first frequency, that frequency being zero in this embodiment. Transmitting-receiving devices of a second type, such as devices 33 and 34, are connected to at least some of the column conductors of the second set of conductors of the network 18 to apply to the network a negative feedback input signal at a second frequency. In this preferred embodiment the first frequency is chosen to be Oc/s, with the advantage that the reference source 12 shown in FIG. 1 can be dispensed with. The second frequency can be chosen to be any value which is known by those skilled in the art to be effective for allowing frequency multiplexing to be carried out effectively between the first and second signals.

Each transmitting-receiving device of the first type comprises a transformer 35 which acts in conjunction with a capacitor 36 to prevent interference between the input and output signals of the device. Thus, D.C output signals are transmitted through the primary winding of transformer 35 while A/C. input signals, after passing through the primary winding of transformer 35, are connected to ground throughcapacitor 36. The secondary winding of the transformer 35 is connected to the input of an AC. amplifier 37 the output of which is connected to the-primary winding of a transformer 38 having a center tap connected through a limiting resistor 39 to an AC. generator 10 connected in common to all the transmitting-receiving devices in the system. The ends of the secondary winding of the transformer 38 are connected to the input of a ring modulator 40 which acts, in actual fact, as a demodulator and is composed of semi-conductor diode devices. The undirectional output signal from the ring modulator 40 is smoothed by a capacitor 41 andapplied to the input of a DC. power amplifier 42 having an"output which can have only a predetermined polarity relative to the source 26. The output of the amplifier 42 is connected to the output 5 of the device 27 through the primary winding of the transformer 35.

At least some of the devices of the first type may include an amplifier 43 which has its input connected to a unidirectional voltage source 44 through a resistance 45 and its output connected to the primary winding of the transformer 35. A feedback loop for the amplifier 43 is provided through a resistance 46. If source 44 is variable or at least one of the resistances 45 and 46 is variable, the amplifier 43, source 44 and resistances 45 and 46 provide a voltage stabilizer circuit which acts, as will be described in more detail hereinafter, as a signal limiting means to define a lowest or highest allowed value for the value of the output signal applied to the conductor 21. The voltage stabilizer circuit can thus enable a problem to be defined more fully in that those variable output signals provided by devices such as device 27 which incorporate a voltage stabilizer circuit can be limited to predetermined allowable values, which values might otherwise be undesirably exceeded or not attained should the stabilizer circuits not be provided. An alternative method of limiting the output signals to the first set of conductors to allowable values would be to provide more columns in the network and correspondingly more transmitting-receiving devices of second type such as devices 33 and 34. However, this latter method would be more expensive than the provision of stabilizer circuits as described and so would not be preferred to such provision.

Each transmitting-receiving device of the second type, for example the devices 33 and 34, has a frequency filter comprising a transformer 47 and a capacitor 48 to separate the incoming and outgoing signals. Incoming zero-frequency signals pass through the secondary winding of the transformer 47 to the input of a DC. amplifier 49. The output of the amplifier 49 is connected through a limiting resistor 50 to a center tap of the secondary winding of a transformer 51, the ends of which winding are connected to the input of a ring modulator 52, the output of which is connected across the input of an A.C. power amplifier 53. The ring modulator 52 is controlled by the amplifier 49 in dependence upon the zero frequency input signal of the device and applies a modulated signal to the primary winding of the transformer 47 through the power amplifier 53. It can be seen that the, combination of the modulators 40 and 52 of all devices such as 27, 33 and 34, together with alternating current generator 20, constitute frequency multiplexing means for the FIG. 2 system. A signal at the second frequency is induced in the secondary winding of the transformer 47, which signal is applied to column conductor 22 of the network 18. The transmitting-receiving devices 33 and 34 are substantially the same except that the amplifier 49 of the device 33 is connected to limiting resistor 50 through a diode 54. Thus, considering the device 34, if an equation is set up on the column 23, when the corresponding equation is satisfied the input signal to the device 34 is equal to zero, so that in practice only a small signal appears at the output of the amplifier 49. When the equation is no longer satisfied, however, a significant output signal is produced by the amplifier 49 which is sufficient to illuminate a lamp 55 and causes an output signal to be applied to terminal 22 by the ring-modulator 52. Lighting of the lamp 55 indicates the lack of feasible solution to the equation, which can also be detected by a relatively large reading on the meter 31. 4

Considering the device 33 however, the output signal of the amplifier 49 applied to the resistor 50 will be zero even when the polarity of the input voltage to the amplifier 49 from the transformer 47 is negative. When the polarity changes to positive, a signal will be applied to the ring-modulator 52 through diode 54 and resistor 50, an output signal will be produced by the device 33 and the system will be out of balance. This out-ofbalance state will be indicated by a lamp 56 and meter 30. Thus, the particular arrangement of the device 33 allows the system to be stable when the zero-frequency signal appearing at the conductor 22 is any negative value. This allows an inequality to be set up on the column associated with conductor 22, the condition of the inequality being that the algebraic sum of its terms must always be equal to or less than zero. It will be appreciated that reversal of the diode 54 will alter the condition of the inequality of being that in which the algebraic sum of its terms must always be equal to or greater than zero.

Thus, stability of the system will be achieved when all the conductors of the second set have such zero frequency signal values produced thereat that the output signals of the transmitting-receiving devices connected to that second set of conductors are substantially equal to zero. It should be appreciated that amplifiers 42 and 49 and the corresponding amplifiers in the other transmitting-receiving devices all have to be carefully determined frequency response in that their response times should be carefully chosen, to assure the stability of the system.

It will be appreciated by those skilled in the art that there is always a problem associated with achieving a stable negative feedback system. The applicant, when building the system described in FIG. 2, found it advantageous to employ as amplifiers 42 and 49, and indeed as all the other amplifiers in this system those manufactured by SGS Fairchild Limited under the designation ,1. A 207. Of the two equivalent types p. A 207A and u A 207C the applicant used the latter because of its relatively low cost as compared with the former. The pr A 207 amplifier is particularly advantageous for use in feedback systems because of its facility for the connection thereto of frequency compensating devices. The need for such devices in any I particular negative feedback system can be determined by the well-known prior art method of testing system stability known as open-loop frequency response testing. Any discussion of'this method and the compensation used in dependence upon its results forms no part set of generators has its own A.C. generator 10 and 12 i for the modulators 9 and 11 while in FIG. 2 the ring modulators 40 and 52 are all fed with the same reference signals of the same frequency from the single A.C. generator 10. The output signals of the transmitting-receiving devices such as 33 and 34 all have the same frequency as the generator 10. This provides that no separate rectification of the modulating signal is required for efficient control of the ring modulators.

The meters 28 to 32 can display values of the output and/or input signals of the transmitting-receiving devices. They are of the edgwise type, the axes of rotation of the pointers of the group of meters such as 28 and 29 being aligned with one another, as are the axes of the pointers of the meters 30 to 32. The meters 30 to 32, for example, if responsive to DC. signals only, would indicate the values of the input signals to devices of the second type, such as 33 and 34, connected to the second set of network conductors. If meters 30 to 32 were responsive to A.C. signals only, they would indicate the output signals of those devices of the second type. Similar considerations apply to the group composed of meters such as 28 and 29. The meter 29 of course would be sensitive to DC. only since any A.C. signal at terminal 20 would have no effect on the stability of the system.

It is to be appreciated that at least some of the impedance elements in the network 18 could be manually variable resistance elements and at least one of the elements could be, for example, a strain gauge attached to a loaded structure. Then, the load applied to this structure would influence the resistance of the strain gauge and cause the system to reach an equilibrium condition dependent upon this load. Again, electrical impedance elements in the system might be replaced by their mechanical, optical, hydraulic or the like equivalents by the provision of transducers as necessary. For example, the link between two give network conductors might comprise an electrically-operated pump driven by the output of the associated voltage generator, an hydraulic pipeline, and a pressure transducer providing an electrical voltage proportional to the pressure transmitted through the pipeline. For this purpose a pump and a transducer would of course be required at each end of the pipeline, unless the hydraulic link were duplicated to allow for signals passing in both directions between the network conductors.

FIG. 3 shows a preferred embodiment of the arrangement of the amplifier 49 of the transmitting-receiving devices 33 and 34 of FIG. 2. In FIG. 3 the output of the amplifier 49 can be connected through one or the other of diodes 57 and 58 to the limiting resistor 50. The input of amplifier 49 is connected to the primary windings of the transformer 47 as in FIG. 2.

If, due the particular condition of an inequality set up at the column conductor 22, stability of the system must be achieved when the input signal to the inverting amplifier 49 is less than zero, then connection is made between terminals A and D and between terminals B and C. Thus, when the input signal is negative no signal is passed to the ring modulator 52 but a signal is passed through the feedback loop comprising diode 58 and a resistor 59. A meter 60 is connected across resistor 59 to give indication of the value of the output of the amplifier 49 from which may be determined what is actually the sum of the terms of the inequality, which sum of course satisfies the condition of the inequality. Conversely, if the condition of the inequality provides that the input signalto the amplifier 49 must be greater than zero for the inequality to be satisfied, then terminals A and C are connected together, as are terminals B and D. The meter 60 gives an indication of the actual value of the inequality as before. If a transmitting-receiving device incorporating the arrangement of FIG. 3 is required to monitor an approximate equation, i.e. to give zero output signal for an output signal which is approximately zero, then all the terminals A, B, C and D are connected to each other and the value of resistor 59 is appropriately chosen.

The gain of the amplifier 49 is reduced by the feedback loop including the resistor 59 so that any small signal, either positive or negative, at the input of the amplifier 49 will produce no significant output signal. A zero output signal of the device 33 could thereby be maintained for small excursions of the input signal about zero volts.

Finally, by connecting A and B to C and leaving D unconnected, the condition of an equality can be sensed Thus the amplifier 49 of a device such as 34 would be connected in this fashion.

FIG. 4 shows a preferred embodiment of the stabilizer circuit of the device 27 shown in FIG. 2 and comprising the battery 44, resistors 45 and 46 and amplifier 43. Switches 61 a and b and 62 a and b connect respective ones of two oppositely poled diodes at the output of amplifier 43 to oppositely poled diodes at the output of amplifier 42. When switches 61 a and b are closed and switches 62 a and b are open it is possible to set a lower limit on the output signal of amplifier 42 which signal is the output of device 27. Conversely, if switches 62 a and b are closed and switches 61 a and b are open, an upper limit will be set on the output signal of device 27 The facility provided by the stabilizer is useful for defining problems in which a solution must attain at least a first determined value or must not exceed a second predetermined value.

FIG. shows a preferred embodiment of a negative feedback system employing frequency multiplexing. A network 18 has a plurality of column conductors three of which are shown at 63, 64 and 74, to each of respective ones of which are connected resistors representing terms of a function, and a plurality of row conductors, some of which are shown at 65, 66 'and68. Transmitting-receiving devices (not shown) are connected to column conductors 63 and 64 and to row conductors 65 and 66, as in the systems shown in FIG. 2. A transmitting-receiving device which does not multiplex between its output an input signals takes the form of an amplifier 67 which has a negative gain and is connected to a row conductor 68 connected to resistors representing constant terms of the functions. I i

The conductor 68 is connected through a resistor 69 to an additional column conductor 70 which applies to the input of a further transmitting-receiving device 71, similar to device 34, a D.C. signal representing the difference between the output signal of the amplifier 67 and the output signal of a variable, unidirectional reference source 72 connected to conductor 70 through a resistor 73. The internal circuitry of the device 71 is identical to that of each of the transmitting-receiving devices such as 34 in FIG. 2 except that its A.C. output is obtained from an additional secondary winding of the transformer 47, which additional winding is illustrated in the block designated 71 in FIG. 5. Device 71 applies to column conductor 74 an A.C. error signal in dependence upon any difference signal on conductor 70, which error signal is applied to network row conductors 65 and 66 through resistors 76 and 77. Conductor 74 and resistor and the additional secondary winding also provide a D.C. negative feedback loop for amplifier 67 so that any D.C. error signal on conductor 74 will result in D.C. negative feedback to the amplifier 67 tending to drive its output signal to equal a predetermined function of the signals on conductors 65 and 66. Said D.C. negative feedback loop is unaffected by the A.C. signal applied by device 71 to conductor 74 since that A.C. signal is connected via a capacitor to ground so as not to appear at the input of amplifier 67.

Consider now, as a simple example of a problem to be solved, the case in which it is desired to produce a clay mix of weight x, tons having constituents a & b in the ratio pl/ql from two clays with constituents a & b in the ratios p2/q2 and p3/q3. It is desired to find the required weights x and x of said two clays.

In the network, the negative output of amplifier 67 is to represent x,, and the voltages applied to conductors 65 and 66 represent the weights x and x respectively. Accordingly the conductances of resistances 78 and 79 are in the ratio pl/ql, the conductances of resistances 80 and 81 are in the ratio p2/q2 and the conductances of resistances 82 and 83 are in the ratio p3/q3. Moreover, x must equal the sum of x and x and this function (equality) is set up by equal value resistances 75, 76 and 77. Any error in this function appears on column conductor 74 and is fed back to the input of amplifier 67. If there is any other error arising from the values x,, x and x supplied a D.C. error signal will be set up on at least one of conductors 63 and 64 and be fed back as an A.C. signal on the appropriate conductor 63 and/or 64. The source 72 provides a signal defining the desired final tonnage, i.e., it defines the desired value of x,. When the signal on conductor 68, representing the feasible value of x,, does not equal the desired value of x,, resistors 69 and 73 set up a D.C. error signal on conductor 70 to actuate the device 71 to create an appropriate A.C. feedback signal on conductor 74 to modify the outputs of the transmitting-receiving devices connected to conductors 65 & 66.

When the system balances, the output of amplifier 67 will be substantially equal to the output of source 72, no appreciable error signal being emitted by device 71. The amounts (tons) of clays x and x required to produce the desired mix x, can be read directly off meters connected to row conductors 65 and 66 respective- It may be that the amount of one of clays x and x 3 required is not available. The amounts read off the meters would then represent an infeasible solution. It will then be necessary to reduce the signal supplied by source 67 until a feasible solution is indicated. Preferably, however, the devices feeding conductors 65 and 66 are in accordance with FIG. 4 and the appropriate maximum values for x and x are set at those devices so that their outputs do not represent infeasible values. Accordingly, when in that case, an insoluble problem has been set, infeasibility will be indicated by illumination of the lamp of device 71 (which is substantially the same as device 34 of FIG. 2) as the output of device 67 will not equal the output of source 72. The source 72 can then be adjusted to extinguish the lamp to give a feasible solution.

On the other hand, if a solution is obtained immediately on switching system on, then the desired value set by source 72 could be increased to search for the maximum obtainable tonnage in the circumstances.

FIG. 6 shows an embodiment of a negative feedback system comprising the network 18, the system in this case employing time multiplexing. D.C. transmittingreceiving devices, such as devices 84 and 85, are connected to conductors of the first and second sets of the network; in addition a constant signal source is connected to one of the conductors of the first set as in previous embodiments. Only devices 84 and 85 are shown for the sake of clarity.

In this embodiment, the signals passing through the network from the first set of terminals to the second set are distinguished from signals passing in the other direction through the network by the fact that the two sets of signals are passed during alternate, distinct, periods of time. This enables both sets of signals to be generated at the same frequency, i.e., zero frequency in the present example. The device 84 comprises an amplifier 86, the output of which amplifier is connected to an electronic, semiconductor, switching device 87 which is shown as a mechanical switch for the sake of simplicity. An amplifier 88 has its input connected to the switching device 87 and its output connected in common with the input of amplifier 86 to one conductor 21 of the network 18. A capacitor 102, acting as dynamic signal storage means, is connected to the switching device 87. The transmitting-receiving device 85 is similar to device 84 and comprises amplifiers 89- and 90, corresponding to amplifiers 88 and 86 respectively, a signal switching device 91 and a storage capacitor 92. A pulse train generator in the form of a clock 93 has two outputs 94 and 95 which can supply pulse trains 4), and to the switches 87 and 91. The pulses of the train (1), occur, in time, in between and distinct from pulses of the train (11 Let a set of linear functions be represented on columns of the network 18. In 4), time the switches 87 and 91 are put into such states that signals pass through the network from the first set of conductors to the second, and in time in such states that signals pass through the network in the opposite direction. Thus the two sets of signals therefore do not interfere with one another even though they are both of zero frequency. It can be seen that the switches 87 and 91 of all devices such as 84 and 85, together with the clock 93, constitute time multiplexing means for the system shown in FIG. 6. The system will reach equilibrium when, in 4), time, very small signals appear at all conductors 22, 23, 24 etc.

The signals appearing at conductors 21, 20, 19 etc. in (1), time will then represent solutions of the linear functions represented on the network.

It will be appreciated that in a time multiplexing negative feedback system such as is shown in FIG. 6 it will be possible to incorporate into the transmittingreceiving devices arrangements similar to those described with reference to FIGS. 3 and 4 of the accompanying drawings. A system employing time multiplexing can thus be designed which will determine equations and/or inequalities and within which limits may beset on values of solutions.

FIG. 7 shows the preferred embodiment of FIG. 5 modified so as to employ time multiplexing according to FIG. 6. A transmitting-receiving device 96 replaces the device 71 of FIG. 5, the device 96 being similar to device of FIG. 6 but having its input and output separated.

Further transmitting-receiving devices (not shown) similar to devices 84 and 85 are connected to conductors such as 63 to 66 of the network. A switch 97 is connected between source 72 and conductor 70 and a switch 98 is connected between conductor 74 and amplifier 67. In (it, time switches 97 and 98 are closed, device 96 and the devices connected to conductors 63 and 64 are switched to receive and the devices connected to conductors 65 and 66 are switched to transmit. In (1) time, switches 97 and 98 are open, and the transmitting-receiving devices are switched to their opposite function.

FIG. 8 shows an-embodiment of a negative feedback system comprising two networks 99 and 100 each having transmitting-receiving devices connected to its first and second sets of conductors according to any of the preceding embodiments.

A first set of linear functions can be set up on network 99 and a second set of linear functions can be set up on network 100. Each of the first and second sets of functions could be solved independently of the other but for the connection from conductors of the first sets of conductors of networks 99 and 100 to a conductor of a first set of terminals of a further network 101. Thus, to achieve balance, a third set of linear functions must be solved having solutions which not only satisfy the third set of functions, but which satisfy the first and second sets of functions also.

It is to be appreciated that each of the networks 99, 100 and 101 can be in the form of network 18 shown in FIG. 2 or of the preferred form shown in FIGS. 5 and 7. In addition, the transmitting-receiving devices associated with the networks 99, 100 and 101 may all be of the frequency multiplexing type shown in detail in FIG. 2 or may all be of the time multiplexing type shown in FIG. 6.

The system shown in FIG. 8 is, of course, relatively simple. Any number and combination of networks could rely on one another for their solutions, and many different relationships between these solutions could be accommodated by arranging the connections between the networks accordingly.

It is to be appreciated that the networks, such as network 18, described in the foregoing embodiments do not represent the full size of the resistance matrix in an analog computer. For example, the network described with respect to FIGS. 5 and-7 might consist of smoothly or step wise variable resistances of a 50 X 50 matrix composed of 2,500 variable resistances interconnected by 50 row and 50 column conductors. These resistances can be set to infinity as may be necessary to implement some of the functions and to provide a conductor such as the row conductor connected to source 72 in FIG. 5 where only one of the resistances connected to the conductor, i.e., resistances 73, will have a finite value.

lclaim:

1. An analog computer comprising an electrical network having a first set of conductors, a second set of conductors and electrical impedance elements coupling said first set of conductors to said second set of conductors such that a signal applied at any one of said conductors of one of said sets will produce at at least one of said conductors of the other of said sets signals having values determined by those of said impedance elements which couple said one conductor to said other set, said analog computer including the improvement which comprises: a negative feedback system including:

a plurality of transmitting-receiving devices each having an output and an input coupled in common to an associated one of said conductors of said first and second sets and each being operable to apply to its output and output signal varying in dependence upon an input signal received at its input; and

multiplexingmeans for multiplexing the signals which pass from said first set of conductors to said second set of conductors with the signals which pass from said second set'of conductors to said first set of conductors.

2. An analog computer as claimed in claim 1,

wherein said multiplexing means are frequency multiplexing means including frequency conversion means for providing said signals which pass from said first set of conductors to said second set of conductors at a first frequency and said signals which pass from said second set of conductors to said first set of conductors at a second frequency.

3. An analog computer as claimed in claim 2, wherein said multiplexing means includes means permitting said first frequency to be zero.

4. An analog computer as claimed in claim 1, and comprising signal generating means connected to at least one of said conductors of said first set and operable to apply to said electrical network and output signal independent of the other signals in said network.

5. An analog computer as claimed in claim 2, wherein said frequency conversion means include modulating means.

6. An analog computer as claimed in claim 3, wherein said frequency conversion means providing those of said signals which are at zero frequency comprise de-modulating means and said frequency conversion mean providing said signals at said second frequency include modulating means.

7. An analog computer as claimed in claim 5 wherein said modulating means are ring modulators, each of which comprises a transformer and a plurality of semiconductor devices. 7

8. An analog computer as claimed in claim 1 wherein said multiplexing means are time multiplexing means arranged to render those of said transmitting-receiving devices. connected to said first set of conductors alternately transmissive and receptive during alternating and distinct first and second periods of time and to render those of said transmitting-receiving devices connected to said second set of conductors alternately receptive and transmissive during said first and second periods of time.

9. An analog computer as claimed in claim 8, wherein said time multiplexing means comprises a clock pulse generator which defines said first and second periods of time.

10. An analog computer as claimed in claim 8 wherein each of said transmitting-receiving devices comprises signal storage means connected to both the input and the output of said transmitting-receiving device.

11. An analog computer as claimed in claim 10 wherein said signal storage means is a capacitance.

12. An analog computer as claimed in claim 1 wherein at least some of said impedance elements are variable value elements.

13. An analog computer as claimed in claim 12 wherein at least one of said variable value elements is a transducer arranged to be sensitive to an influence external of said computer.

14. An analog computer as claimed in claim 1 wherein at least one of those of said transmittingreceiving devices which are coupled to said first set of conductors comprises signal limiting means operable to limit the possible range of values of its output signal to a predetermined maximum value.

15. An analog computer as claimed in claim 1 wherein at least one of those of said transmittingreceiving devices which are coupled to said first set of conductors comprises signal limiting means operable to limit the possible range of values of its output signal to a predetermined minimum value.

16. An analog computer as claimed in claim 1 wherein at least one of those of said transmittingreceiving devices which are coupled to said second set of conductors comprises diode means coupled between its input and output so poled as to provide a substantially zero output signal for the device when its input signal is of a predetermined polarity.

17. An analog computer as claimed in claim 4 and comprising a further transmitting-receiving device which is coupled to a further conductor of said first set of conductors and is operable to produce an output signal in dependence upon an input signal which is an output signal at one of said conductors of said second set of conductors and comparison means having an input coupled to the outputs of said signal generating means and said further transmitting-receiving device and having an output coupled to a conductor of said second set.

18. An analog computer as claimed in claim 17, wherein said comparison means is an additional multiplexed transmitting-receiving device having an input coupled to a conductor of said second set of conductors, and having an output coupled to a conductor of said second set of conductors.

19. An analog computer as claimed in claim 18, wherein said output of said additional transmittingreceiving device is connected to said one conductor of said second set.

20. An analog computer as claimed in claim 1, further comprising a plurality of signal indicating means connected to at least some of said conductors of at least one of said first and second sets for providing indication of at least one of the input and output signals appearing at said at least some conductors.

21. An analog computer for computing optimum values for satisfying a set of simultaneous relations, said analog computer comprising an electrical network having a first set of conductors, a second set of conductors, and electrical impedance elements coupling said first set of conductors to said second set of conductors, said set of simultaneous relations being set up on said electrical network and said electrical network being such that a signal applied at any one of said conductors of one of said sets will produce at at least one of said conductors of the other of said sets signals having values determined by those of said impedance elements which couple said one conductor to said other set, said analog computer including the improvement which comprises; a negative-feedback system including;

first and second sets of transmitting-receiving devices, each of said devices having output-input means and being operable to produce at its outputinput means an output signal having a value dependent upon the value of an input signal received by said device at its output input means, each device of said first set of devices having its output-input means coupled to an associated one of said conductors of said first set of conductors and each device of said second set of devices having its output-input means coupled to an associated one of said conductors of said second set of conductors, said devices of said first set being operable to supply to said first set of conductors output signals which will influence through said network said second set of devices to cause said second set of devices to supply negative feedback output signals back through said network to said first set of transmitting-receiving devices, thereby to drive the computer towards an equilibrium condition in which said first set of devices supplies to said network solutions to said set of simultaneous relations; and

multiplexing means for multiplexing the signals which pass from said first set of transmittingreceiving devices through said network with the negative feedback signals which pass from said second set of transmitting-receiving devices back through said network. 

1. An analog computer comprising an electrical network having a first set of conductors, a second set of conductors and electrical impedance elements coupling said first set of conductors to said second set of conductors such that a signal applied at any one of said conductors of one of said sets will produce at at least one of said conductors of the other of said sets signals having values determined by those of said impedance elements which couple said one conductor to said other set, said analog computer including the improvement which comprises: a negative feedback system including: a plurality of transmitting-receiving devices each having an output and an input coupled in common to an associated one of said conductors of said first and second sets and each being operable to apply to its output and output signal varying in dependence upon an input signal received at its input; and multiplexing means for multiplexing the signals which pass from said first set of conductors to said second set of conductors with the signals which pass from said second set of conductors to said first set of conductors.
 1. An analog computer comprising an electrical network having a first set of conductors, a second set of conductors and electrical impedance elements coupling said first set of conductors to said second set of conductors such that a signal applied at any one of said conductors of one of said sets will produce at at least one of said conductors of the other of said sets signals having values determined by those of said impedance elements which couple said one conductor to said other set, said analog computer including the improvement which comprises: a negative feedback system including: a plurality of transmitting-receiving devices each having an output and an input coupled in common to an associated one of said conductors of said first and second sets and each being operable to apply to its output and output signal varying in dependence upon an input signal received at its input; and multiplexing means for multiplexing the signals which pass from said first set of conductors to said second set of conductors with the signals which pass from said second set of conductors to said first set of conductors.
 2. An analog computer as claimed in claim 1, wherein said multiplexing means are frequency multiplexing means including frequency conversion means for providing said signals which pass from said first set of conductors to said second set of conductors at a first frequency and said signals which pass from said second set of conductors to said first set of conductors at a second frequency.
 3. An analog computer as claimed in claim 2, wherein said multiplexing means includes means permitting said first frequency to be zero.
 4. An analog computer as claimed in claim 1, and comprising signal generating means connected to at least one of said conductors of said first set and operable to apply to said electrical network and output signal independent Of the other signals in said network.
 5. An analog computer as claimed in claim 2, wherein said frequency conversion means include modulating means.
 6. An analog computer as claimed in claim 3, wherein said frequency conversion means providing those of said signals which are at zero frequency comprise de-modulating means and said frequency conversion mean providing said signals at said second frequency include modulating means.
 7. An analog computer as claimed in claim 5 wherein said modulating means are ring modulators, each of which comprises a transformer and a plurality of semiconductor devices.
 8. An analog computer as claimed in claim 1 wherein said multiplexing means are time multiplexing means arranged to render those of said transmitting-receiving devices connected to said first set of conductors alternately transmissive and receptive during alternating and distinct first and second periods of time and to render those of said transmitting-receiving devices connected to said second set of conductors alternately receptive and transmissive during said first and second periods of time.
 9. An analog computer as claimed in claim 8, wherein said time multiplexing means comprises a clock pulse generator which defines said first and second periods of time.
 10. An analog computer as claimed in claim 8 wherein each of said transmitting-receiving devices comprises signal storage means connected to both the input and the output of said transmitting-receiving device.
 11. An analog computer as claimed in claim 10 wherein said signal storage means is a capacitance.
 12. An analog computer as claimed in claim 1 wherein at least some of said impedance elements are variable value elements.
 13. An analog computer as claimed in claim 12 wherein at least one of said variable value elements is a transducer arranged to be sensitive to an influence external of said computer.
 14. An analog computer as claimed in claim 1 wherein at least one of those of said transmitting-receiving devices which are coupled to said first set of conductors comprises signal limiting means operable to limit the possible range of values of its output signal to a predetermined maximum value.
 15. An analog computer as claimed in claim 1 wherein at least one of those of said transmitting-receiving devices which are coupled to said first set of conductors comprises signal limiting means operable to limit the possible range of values of its output signal to a predetermined minimum value.
 16. An analog computer as claimed in claim 1 wherein at least one of those of said transmitting-receiving devices which are coupled to said second set of conductors comprises diode means coupled between its input and output so poled as to provide a substantially zero output signal for the device when its input signal is of a predetermined polarity.
 17. An analog computer as claimed in claim 4 and comprising a further transmitting-receiving device which is coupled to a further conductor of said first set of conductors and is operable to produce an output signal in dependence upon an input signal which is an output signal at one of said conductors of said second set of conductors and comparison means having an input coupled to the outputs of said signal generating means and said further transmitting-receiving device and having an output coupled to a conductor of said second set.
 18. An analog computer as claimed in claim 17, wherein said comparison means is an additional multiplexed transmitting-receiving device having an input coupled to a conductor of said second set of conductors, and having an output coupled to a conductor of said second set of conductors.
 19. An analog computer as claimed in claim 18, wherein said output of said additional transmitting-receiving device is connected to said one conductor of said second set.
 20. An analog computer as claimed in claim 1, further comprising a plurality of signal indicating means connected to at least some of said conductors of at least one of said first and second sets for providing indication of at least one of the input and output signals appearing at said at least some conductors. 